After graduating high school I worked at a local Ford dealership as a mechanic, specializing in manual transmissions and differentials. I've pulled and reinstalled more than a couple of sets of axle shafts! I've been following your thread for a bit now and closely watching all of your comments as well as examining your pictures closely. I didn't want to post any suggestions too quickly until I had what I felt was a pretty complete picture of your situation. This is what I've been able to see; your replacement axle seal appears to have been seated correctly. If it fit over the seal surface on the axle shaft, it was the correct one to begin with. Many modern seals will not look exactly as the original stuff. What matters is outer diameter, inner (sealing surface) diameter, depth and seal type (e.g. double lip or single lip seals). The appearance of the carrier splines seeming to be out of alignment is not possible. If an incorrect carrier bearing was installed it would not fit over the end of the carrier properly, nor would it sit into the bearing saddle properly. If the bearing is fitting both surfaces properly (which it appears to be), the splines of the carrier will be in proper alignment.
Based on everything I've read from your posts and pictures/video I'd have to come to one conclusion, you aren't getting the spline end of the axle into the carrier opening, plain and simple. Based on the pictures you posted with the axle going in without the seal, but not with the seal installed shows it is engaging properly. It's also seated at the proper depth in the " no seal installed" photo. Based on the lower photo and the comments you made about putting the bolts into the retainer plate causing the axle to lock down further tells me misalignment of the shaft at the carrier end. With the seal out you have the entire diameter of the axle housing tube in which to move the shaft around to align it for engagement. When you have the seal installed there isn't much room to compensate for error. What appears is happening is you're poking it into the area of the adjustment spanner nut or against the end of the ring gear carrier. When you installed the bolts to try to draw the shaft in you were just lock the tip of the axle against the spanner nut. Remember, when you have the shaft most of the way in and start moving the flange to align it, a little movement on the axle flange translates into a large amount of movement at the spline end. Sometimes they can be a total bear to seat!
I'd suggest putting everything back together and ready to accept the axle shafts. Insert the axle shaft into the housing, supporting the center with one hand and the other hand on the flange. Once you have it in approximately 2/3's of the way, grab the axle flange with both hands, one hand opposite the other on the flange. Slide the shaft into the housing keeping the shaft as level and straight as you can, as far as you can. As soon as you feel contact with anything, gently push a little harder. Also, turn the shaft back and forth slightly to allow for spline alignment as you push. Keep in mind there is a smaller taper to the opening of the ring gear carrier at the axle shaft opening. This helps a little in alignment. If the edge of the shaft is against that taper, the additional pressure will force the shaft into the opening in proper alignment. You'll also see the axle shaft bearing going deeper into the axle housing. It may then take a few gentle taps with a hammer on the flange to seat it fully. If it DIDN'T hit the taper, you're still against the end of the ring gear carrier end. Wiggle it GENTLY and in SMALL movements. You should be able to feel it when it falls into alignment. At that point PUSH, you don't want it dropping back out!! Also keep in mind when you're moving it around holding just the axle flange, you don't have a whole lot of leverage. I usually make the up/down or side to side corrections with my finger tips while countering with pressure on the heels of my palms for leverage (I hope that's a clear explanation of that point!). As I said, some go in easily (luck!), and some are a total PITA! One final suggestion, once you start the axle into the housing while holding the axle flange, close your eyes. Just concentrate on what you're feeling and not what you're seeing. I'm not screwing with you, I'm totally serious. It truly works! Good luck with it and keep us all posted!